nLab Eilenberg-MacLane space type



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels





Let GG be an abelian group, and nn a natural number. The Eilenberg-MacLane space K(G,n)K(G,n) is the unique space that has π n(K(G,n))G\pi_n(K(G,n))\cong G and its other homotopy groups trivial. These spaces are a basic tool in classical algebraic topology, they can be used to define cohomology.

There is an analogous construction in homotopy type theory.


Let GG be a group. The Eilenberg-MacLane space type K(G,1)K(G,1) is the higher inductive 1-type with the following constructors:

  • base:K(G,1)base : K(G,1)
  • loop:G(base=base)loop : G \to (base = base)
  • id:loop(e)=refl baseid : loop(e) = refl_{base}
  • comp: (x,y:G)loop(xy)=loop(y)loop(x)comp : \prod_{(x,y:G)} loop(x \cdot y) = loop(y) \circ loop(x)

basebase is a point of K(G,1)K(G,1), looploop is a function that constructs a path from basebase to basebase for each element of GG. idid says the path constructed from the identity element is the trivial loop. Finally, compcomp says that the path constructed from group multiplication of elements is the concatenation of paths.

It should be noted that this type is 1-truncated which could be added as another constructor:

  • x,y:K(G,1) p,q:x=y r,s:p=qr=s\displaystyle \prod_{x,y:K(G,1)} \prod_{p,q:x=y} \prod_{r,s:p=q} r = s

Recursion principle

To define a function f:K(G,1)Cf : K(G,1) \to C for some type CC, it suffices to give

  • a point c:Cc : C
  • a family of loops l:G(c=c)l : G \to (c = c)
  • a path l(e)=idl(e)=id
  • a path l(xy)=l(y)l(x)l(x \cdot y) = l(y) \circ l(x)
  • a proof that CC is a 1-type.

Then ff satisfies the equations

f(base)cf(base) \equiv c
ap f(loop(x))=l(x)ap_{f} (loop(x))=l(x)

It should be noted that the above can be compressed, to specify a function f:K(G,1)Cf : K(G,1) \to C, it suffices to give:

  • a proof that CC is a 1-type
  • a point c:Cc : C
  • a group homomorphism from GG to Ω(C,c)\Omega(C,c)

See also


Formal proof that the K(G,)K(G,-) form a spectrum:

Last revised on January 31, 2024 at 09:28:09. See the history of this page for a list of all contributions to it.